Defintion: Let $V$ be a normed linear space. A sequence $\{v_i\}_{i=1} ^ \infty \subseteq V$ is said to converge to $v \in V$ weakly if, for each bounded linear functional $T:V \to \mathbb{R}$, we have $\lim_{n \to \infty} T(v_n)=T(v)$.
Q: If $\{v_n\}_{i=1} ^ \infty$ converges weakly to both $v$ and $w$, does this imply $v=w$?
I know that this is true for $L^p(E)$ spaces (at least when $1 \leq p < \infty$), but the proof that I am aware of uses a rather ad hoc method so I'm not convinced that it's true in general.
Best Answer
Yes, it is valid for any normed linear space. If $x^*(x)=0$ for all $x^*\in X^*$, then $x=0$. If $x$ were a nonzero element, the Hahn-Banach extension theorem guarantees a nonzero bounded linear functional $x^*$ such that $x^*(x)=\|x\|\neq0$.